We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded domain. Although this problem is highly nonlinear, we do not require an initial guess of the true solution. The key in our method is the derivation of a boundary value problem for a system of coupled quasilinear elliptic equations whose solution is the vector function of the spatially dependent Fourier coefficients of the solution to the governing parabolic equation. We solve this problem by an iterative method. The global convergence of the system is rigorously established using a Carleman estimate. Numerical examples are presented.

中文翻译：

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一种从横向柯西数据恢复非线性抛物线方程初始条件的收敛数值方法

我们提出了一种新的数值方法，用于通过对有界域边界上的 Dirichlet 和 Neumann 数据的测量来重建拟线性抛物线方程的初始条件的问题。虽然这个问题是高度非线性的，但我们不需要对真实解的初始猜测。我们方法的关键是推导耦合拟线性椭圆方程系统的边值问题，该方程的解是控制抛物线方程解的空间相关傅立叶系数的向量函数。我们通过迭代方法解决这个问题。系统的全局收敛是使用卡尔曼估计严格建立的。给出了数值例子。